Properties of Square - Based on Pattern

Concept Explanation

Properties of Square - Based on Pattern

A pattern refers to a series or sequence which repeats. There are two major kinds of math patterns. They are number patterns or sequences of numbers. These are arranged as per a rule or rules or a formula. The other type we have are shape patterns that are tagged by making use of letters in a definite shape and the way that they repeat.

Some interesting patterns of square numbers :

Pattern 1:The sum of two consecutive triangular numbers is a square number .

Numbers whose dot patterns can be arranged as triangles are known as triangular numbers.1,3,6,10,15,21,... are triangular numbers.This fact can be observed by the following pattern

$1=1^{2}$

$1+3=4=2^{2}$

$3+6=9=3^{2}$

$6+10=16=4^{2}$

$10+15=25=5^{2}$

$15+21=36=6^{2}$ etc.

Also n^th triangular number is given by $frac{n(n+1)}{2}$

Pattern 2: If 1 is added to the product of two consecutive odd natural numbers,it is equal to the square of the only even natural number between them.

This fact can be observed by the following pattern

$1times3+1=4=2^{2}$

$3times5+1=16=4^{2}$

$5times7+1=36=6^{2}$

$7times9+1=64=8^{2}$

$9times11+1=100=(10)^{2}$ etc.

In general, $(2n-1)times(2n+1)+1=4n^{2}=(2n)^{2}$

Pattern 3: If 1 is added to the product of two consecutive even natural numbers,it is equal to the square of the only odd natural number between them.

This fact can be observed by the following pattern

$2times4+1=9=3^{2}$

$4times6+1=25=5^{2}$

$6times8+1=49=7^{2}$

$8times10+1=81=9^{2}$

$10times12+1=121=(11)^{2}$ etc.

In general: $2ntimes(2n+2)+1=4n^{2}+4n+1=(2n+1)^{2}$

Pattern 4:The square of any odd natural number other than 1 can be expressed as the sum of two consecutive natural numbers.

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Sample Questions

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Question : 1

Observe the pattern $2^{2}-1^{2}=2+1$ $3^{2}-2^{2}=3+2$ $4^{2}-3^{2}=4+3$ $5^{2}-4^{2}=5+4$ and find $(111)^{2}-(109)^{2}$
A. 420	B. 430	C. 340	D. 440
Right Option : D
View Explanation

Question : 2

Using the given pattern,find the missing numbers: $1^{2}+2^{2}+2^{2}=3^{2}$ $2^{2}+3^{2}+6^{2}=7^{2}$ $3^{2}+4^{2}+(12)^{2}=13^{2}$ $4^{2}+5^{2}+(.....)^{2}=(21)^{2}$ $5^{2}+(......)^{2}+(30)^{2}=(31)^{2}$ $6^{2}+(7)^{2}+(.....)^{2}=(.....)^{2}$
A. 20,6,43,48	B. 20,6,42,43	C. 21,25,43,42	D. none of these
Right Option : B
View Explanation

Question : 3

Using suitable pattern compute $frac{(666666)^{2}}{12345654321}$
A. 36	B. 32	C. 38	D. 46
Right Option : A
View Explanation

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Rules For Divisibility

Number Theory

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